Freezing sandpiles and Boolean threshold networks: equivalence and complexity
aa r X i v : . [ c s . D M ] J a n Freezing sandpiles and Boolean threshold networks:equivalence and complexity
Eric Goles , Pedro Montealegre , and K´evin Perrot Facultad de Ingenieria y Ciencias, Univ. Adolfo Iba˜nez, Santiago, Chile Aix Marseille Univ., Univ. de Toulon, CNRS, LIS, UMR 7020, Marseille, France Univ. Cˆote d’Azur, CNRS, I3S, UMR 7271, Sophia Antipolis, France
Abstract
The NC versus P -hard classification of the prediction problem for sandpiles on thetwo dimensional grid with von Neumann neighborhood is a famous open problem.In this paper we make two kinds of progresses, by studying its freezing variant.First, it enables to establish strong connections with other well known predictionproblems on networks of threshold Boolean functions such as majority. Second, wecan highlight some necessary and sufficient elements to the dynamical complexity ofsandpiles, with a surprisingly crucial role of cells with two grains. The sand pile model, as well as the Boolean threshold automata, have been studied andapplied extensively in various domains [4, 20, 9, 14, 12, 7, 13]. The classical sandpilemodel on the two dimensional grid with von Neumann neighborhood was introduced inthe 1980 by Bak, Tang and Wiesenfeld, as a simple and natural model of some physicalphenomena [1]. In [8] Goles and Margenstern showed that in arbitrary graphs, any givenTuring machine can be simulated by a configuration of the sandpile model. This meansthat on an arbitrary topology the dynamic of the sandpile model is Turing-universal.After that, in [21], Moore and Nilsson started the study of how difficult it is to predictthe behavior of sandpiles, bringing the question to the formal theory of computationalcomplexity. Sandpile prediction problems are usually solvable in polynomial time bysimply running the simulation until the dynamics reaches a stable state. Essentially,the results of Moore and Nilsson say that sandpiles in one dimension are efficientlypredictable in parallel (in NC ), and that sandpiles in three dimensions or more areintrinsically sequential ( P -complete). It leaves open the two-dimensional case, which hasnot yet been closed despite considerable efforts [6, 5, 13, 24, 3].Following a trend of research leading to new discoveries around well known openproblems on majority dynamical systems (reviewed in Section 3), we introduce in thispaper the freezing variant of sandpiles, where each site can be fired at most once. Very1nterestingly, the freezing world breaks a fundamental barrier between majority and sand-piles. Though it is known that two-dimensional majority can simulate two-dimensionalsandpiles [13], it is unknown whether the converse is true. Indeed, the main difficultylies in the so called abelian property of sandpile models (the fact that sand grains maybe toppled in any order), which is absent in the majority rule. This later model there-fore heavily depends on the parallel schedule of cells update (see [10]), which is not thecase in sandpiles and makes a simulation result hard to establish. It turns out that thefreezing world breaks this frontier, as majority and other threshold Boolean functionsare not sensitive to the order of cells update in this case. Predicting freezing variantsof dynamical systems may be thought as a “simplest case” study of their complexity(see [4, Proposition 6] and Remark 1 for a formal discussion).The complexity classes at stake are AC (constant time in parallel), NL (non-deterministiclogarithmic space), NC (poly-logarithmic time in parallel), and P (polynomial time), with AC ⊆ NL ⊆ NC ⊆ P (see for example [16]).In Section 2 we define the model and problem under consideration and in Section 3 wereview results on the computational complexity of prediction problems in the freezingworld. Section 4 establishes in this setting an isomorphism between the dynamics ofsandpiles and threshold Boolean functions on a grid layout. Finally, Section 5 studies allpossible restrictions for the freezing sandpile prediction problem, consisting in allowingonly a subset of sand contents in the configuration given as input. All but two cases areclassified as being in NC or as hard as the general (freezing) case, which allows to defineeven simpler sandpile prediction problems yet preserving the complexity of the generalcase. The two remaining cases are discussed at the end of the Section; the difficulty torelate them with other models brings novel insights on a possible hierarchy of sandpileprediction problems, between NC and P . We consider the freezing variant of the classical sandpile model introduced by Bak, Tangand Wiesenfled in [1] on the two dimensional grid with von Neumann neighborhood. A configuration c ∈ ( N ∪ {−∞} ) Z assigns a number of sand grains to each cell of the grid,or −∞ when a cell has already fired. For commodity let c v denote the sand content atposition v ∈ Z in configuration c . When the sand content of a cell exceeds its number ofout-neighbors (four in the grid with von Neumann neighborhood), then the cell gives onegrain to each of its out-neighbors and enters the state −∞ ( freezing state ) so that it nevergives grains again. Formally, with N (( i, j )) = { ( i, j + 1) , ( i + 1 , j ) , ( i, j − , ( i − , j ) } the dynamics is defined by F : ( N ∪ {−∞} ) Z → ( N ∪ {−∞} ) Z such that for all v ∈ Z , F ( c ) v = −∞ if c v ≥ c v + P u ∈N ( v ) N ( c u −
4) otherwisewhere N ( x ) is the indicator function of N , which equals 1 when x ≥
0, and 0 when x <
0, for any x ∈ Z . Remark that this discrete dynamical system is deterministic.2hen a cell gives grains to its neighbors we say that it fires , and immediately freezes .Classically, a configuration c is finite when the number of non-empty cells is finite,that is when |{ v | c v = 0 }| < ∞ . Note that up to translation, all non-empty cells of afinite configuration can always be placed inside a rectangle of (finite) size n × m withthe bottom left corner at the origin, hence going from (0 ,
0) to ( n − , m − finite configuration c ashaving the freezing state outside the encompassing rectangle, that is with c ( v x ,v y ) = −∞ when v x < v x ≥ n or v y < v y ≥ m . A configuration c is stable when no grainmoves, that is when c v < v . Additionally, we say that a finite configuration c is simple when for all cell v inside the rectangle of size n × m we have c v ∈ { , , , , } . Freezing sandpiles prediction problem (FSPP)
Input: a simple finite configuration c and a cell v . Question: does there exist t such that F t ( c ) v ≥ FSPP ∈ P ) by running the simulation: one step takes a linear time to be computed, and atleast one cell freezes at each step or we have reached a stable configuration. Since cellsremain frozen, after linearly many steps the configuration is stable, and we can answer.Let Sandpiles prediction problem ( SPP ) be the analogous prediction problemon classical (non-freezing) sandpile model, the one not known to be in NC nor P -hard.We denote ≤ m NC the many-one reduction in NC , and ≤ m AC the many-one reduction in AC . Remark 1.
As stated in [24] (Lemma 1), when there is only one value , and if fur-thermore this value is placed on the border of the n × m rectangle containing the finiteconfiguration, then SPP is the same as
FSPP because each cell is fired at most once.
For the sandpile model [21] and majority cellular automata [20], it has early been proventhat the prediction problem is in NC for dimension one, and P -hard for dimension threeand above. Such P -hardness results [18, 22, 23] are commonly proven via reductionsfrom the canonical circuit value problem (CVP) originally proven to be P -complete byLadner [17], or its monotone variant (MCVP), or its planar variant (PCVP) (see [16]).It is remarkable that all these reductions employ Bank’s encoding [2]: dynamical chainsof reactions implement circuit computations on a quiescent background, as electronsmoving along wires. Note that planar monotone circuit value problem (PMCVP), whichis easily reducible to sandpiles and majority prediction problems (even freezing), hashowever been proven to be in NC , i.e. efficiently computable in parallel [27].Studies of the freezing world have been introduced in cellular automata [15], wherethe authors prove that Turing universality can be achieved even in one dimension, and3hat the prediction problem may be P -complete in two dimensions, though in one di-mension it is in NL . On the two-dimensional grid with von Neumann neighborhood, itis proven in [3] that at least two state changes are necessary to be intrinsically universalaccording to block simulation, and furthermore that two state changes are sufficient. Thepaper [25] places freezing cellular automata universality and prediction in the broadercontext of bounded-change and convergent cellular automata.Previous works on the prediction of threshold (majority-like) functions will be usefulin our analysis of sandpile prediction problems complexity. It is known that predicting freezing strict majority is P -complete for undirected graphs of maximum degree at leastfive (∆( G ) ≥ NC for undirected graphs of maximum degree at most four(∆( G ) ≤
4) [14]. Regarding freezing non-strict majority , its prediction is P -complete forundirected graphs of maximum degree at least four (∆( G ) ≥ NC for undirectedgraphs of maximum degree at most three (∆( G ) ≤
3) [14]. This latter has also beenproven to be in NC for the two-dimensional grid with von Neumann neighborhood [7].Remark that, although planarity is known to forbid information crossing on sandpilemodels [5, 24], it is an obstacle that can be overcome on non-freezing majority [11](based on a planar traffic light gadget of degree five, exploiting the non-freeziness). We begin with a remark linking the dynamics of (freezing) sandpiles to that of anassembly of Boolean functions. These relations will be useful to employ the literature inorder to prove that some problems are in NC (Section 5). Indeed, the dynamics of finitefreezing sandpiles can be seen as a grid network of freezing threshold Boolean functions.Let us define finite freezing Boolean networks on the two dimensional grid with vonNeumann neighborhood. Let G n × m = ( V n × m , E n × m ) be the finite undirected graphdefined as the subgraph of the two dimensional grid induced by vertices in the rectangleof size n × m with the bottom left corner at the origin. Formally, V n × m = { ( x, y ) ∈ Z | ≤ x ≤ n − ≤ y ≤ n − } E n × m = { ( u, v ) ∈ V | v ∈ N ( u ) } For simplicity, when the dimensions are clear from the context, we will denote G = ( V, E )such a graph. The set of configurations is { , } V , and each vertex v is equipped with alocal Boolean function which is freezing (state 1 is always sent to state 1). We use fivesuch local functions, given a configuration c : • ∧ ( and ) defined as f ∧ v ( c ) = c v = 1 or P u ∈N ( v ) c u = 40 otherwise • M ( strict majority ) defined as f Mv ( c ) = c v = 1 or P u ∈N ( v ) c u >
20 otherwise4 m ( non-strict majority ) defined as f mv ( c ) = c v = 1 or P u ∈N ( v ) c u ≥
20 otherwise • ∨ ( or ) defined as f ∨ v ( c ) = c v = 1 or P u ∈N ( v ) c u ≥
10 otherwise • constant
1) defined as f v ( c ) = 1Note that each local function only depends on the state of the vertex and its neigh-bors, and is invariant by permutation of the neighbors, as is the case in freezing sandpiles.Also, the formulation of local functions takes into account the fact that some vertices onthe border of the graph G are missing some neighbors (the number of neighbors, four, ishard-coded in the local functions). Let B : { , } V → { , } V be the dynamics obtainedby applying in parallel the local function assigned to each vertex, i.e. such that for all v ∈ V we have B ( c ) v = f v ( c ) . Given a finite simple sandpile configuration c of size n × m for the freezing sandpilemodel, we define the corresponding freezing threshold Boolean network B c of size n × m (where the local function f v at v depends on the value of c v ) by c v f v ∧ M m ∨ φ ( c ) v = (cid:26) c v = −∞ c of size n × m , we obtain the freezingBoolean network B c of size n × m , which dynamics commutes with the transformation φ on configurations. Proposition 1.
For any finite simple c and all t ∈ N we have B tc ( φ ( c )) = φ ( F t ( c )) .Proof. Note that initially freezed sandpile cells (outside the rectangle of size n × m )are discarded in the corresponding freezing Boolean network. Starting from the initialconfiguration with all cells in Boolean state 0, this latter will therefore simply transforma grain move from u to v into the fact that u is a neighbor of v in state 1. At eachtime step, a cell in the freezing state remains in the freezing state in both dynamics.Regarding other cells, in both dynamics and at each step, they enter the freezing stateif and only if at least the same number (in both dynamics) of their neighbors are in thefreezing state. And φ depends only on the freezing (or not) state of each cell.5 Computational complexity of FSPP
We study the computational complexity of restrictions on
FSPP , depending on the sandcontents that each cell of the simple finite configuration given as input can take among { , , , , } . It is obvious that forbidding the value 4 leads to answering no to anyprediction question, and allowing only the values 3 and 4 (or just 4) leads to always an-swering yes , therefore we consider only the 14 remaining cases. For any A ⊆ { , , , , } we say that a configuration c is A -simple when for all cell v we have c v ∈ A . With thisnotation, simple means { , , , , } -simple. A -freezing sandpiles prediction problem ( A -FSPP) Input: an A -simple finite configuration c and a cell v . Question: does there exist t such that F t ( c ) v ≥ A ⊆ { , , , , } is sound, as follows. Proposition 2.
If we generalize the definition of A -simple configuration, then we stillhave A -FSPP ≤ m AC FSPP for any finite A ⊆ N .Proof. Let ( c, v ) be an instance of A -FSPP . Since the model is freezing, the cell tocell transformation of c into c ′ defined as c v min { c v , } (note that the outer partof the finite rectangle is not modified) preserves the answer (after one step we have F ( c ) = F ( c ′ )), and ( c ′ , v ) is an instance of FSPP .Considering A among { , } , { , } , { , , } , { , } , { , , } , { , , } , { , , , } , { , , } , { , , } , { , , , } , { , , } , { , , , } , { , , , } , { , , , , } , the results are summed-up in Theorems 1 and 2, plus the Open question 1. Theorem 1. A -FSPP ∈ NC when A is one of { , } , { , } , { , , } , { , } , { , , } , { , , } . Theorem 2. FSPP ≤ m AC A -FSPP when A is one of { , , } , { , , } , { , , , } , { , , , } , { , , , } and { , , , , } . Open question 1. { , , , } -FSPP ≤ m AC { , , } -FSPP , but does { , , } -FSPP ∈ NC or FSPP ≤ m AC { , , , } -FSPP ? Subsections 5.1 and 5.2 will present respectively the results of Theorems 1 and 2.Subsection 5.3 will present some perspectives on Open question 1.
This section makes use of the developments presented in Section 4, in order to applyresults from the literature on problems in NC .6 ∞−∞ −∞−∞ −∞−∞ −∞−∞ −∞−∞ −∞−∞ −∞−∞−∞ −∞−∞ −∞−∞ −∞ AC of a { , } -simple sandpile configuration to a configu-ration for the freezing strict majority dynamics on the grid [14]. { , } -FSPP When
FSPP is restricted to { , } -simple configurations, according to Proposition 1 itcorresponds to a finite freezing Boolean network on the grid with only and and constant c, v ) ispositive if and only if φ ( c ) v = 1 or P u ∈N ( v ) φ ( c ) u = 4 (with |N ( v ) | ≤ Proposition 3. { , } -FSPP ∈ AC . { , } -FSPP and { , , } -FSPP When
FSPP is restricted to { , } -simple configurations, according to Proposition 1 itcorresponds to a finite freezing Boolean network on the grid with only strict majority and constant constant strictmajority cells initially in state 1 since we are in a freezing world. As a consequence, we areleft with only strict majority local functions on a grid with von Neumann neighborhood,which can be predicted in NC according to [14] (to adapt the setting it is sufficient toadd a border of cells in state 0). The transformation is easily performed in AC , leadingto an overall algorithm in NC . See Figure 1 for an illustration. Proposition 4. { , } -FSPP ∈ NC . The result of [14] can also be applied to prove that { , , } -FSPP is in NC . Theidea is that cells u with c u = 0 are completely passive in the freezing dynamics (theyfire if and only if all their four neighbors are already fired and frozen). More precisely,given an instance ( c, v ) we consider two cases.1. If c v = 0 then we perform in AC the following modification of the grid: each vertex u = ( u x , u y ) such that c u = 0 is replaced with four vertices u n , u e , u s , u w and thearcs { ( u x , u y + 1) , u n } , { ( u x + 1 , u y ) , u e } , { ( u x , u y − , u s } , { ( u x − , u y ) , u w } , and { u n , u e } , { u e , u s } , { u s , u w } , { u w , u n } . With state 1 on vertices u such that c u = 4and state 0 elsewhere, answers to the prediction under freezing strict majority onthis graph G and to the freezing sandpiles prediction problem are identical. Indeed,in the freezing strict majority dynamics the newly created vertices correspondingto cells such that c u = 0 will never reach state 1 because they always have two oftheir three neighbors in state 0. Since cells such that c u = 0 are completely passive7 ∞−∞ −∞−∞ −∞−∞ −∞−∞ −∞−∞ −∞−∞ −∞−∞−∞ −∞−∞ −∞−∞ −∞ Figure 2: Transformation in AC of a { , , } -simple sandpile configuration to a graph(of maximum degree 4) for the freezing strict majority dynamics [14]. Vertices in state0 (resp. 1) are white (resp. black).in the sandpile dynamics ( i.e. considering that they do not fire leaves the behaviorof other cells unchanged), and since the questioned cell v is not one of these, v willfire from c if and only if it reaches state 1 in the freezing strict majority dynamicson G . Finally, we have ∆( G ) ≤
4, therefore [14] gives an NC algorithm to predictthe freezing strict majority dynamics. See an example on Figure 2.2. If c v = 0 then, if furthermore at least one of the four neighbors of v is 0 (or −∞ ) then v cannot fire and the answer is negative. Otherwise we do the sametransformation as in the case c v = 0, and ask if each of the four neighbors of v willfire (still in NC ). The answer for v is positive (it will fire) if and only if all its fourneighbors will fire. Proposition 5. { , , } -FSPP ∈ NC . { , } -FSPP When
FSPP is restricted to { , } -simple configurations, according to Proposition 1 itcorresponds to a finite freezing Boolean network on the grid with only non-strict majority and constant non-strict majority cells initiallyin state 1 since we are in a freezing world, and can be decided in NC according to [7]. Proposition 6. { , } -FSPP ∈ NC . { , , } -FSPP When
FSPP is restricted to { , , } -simple configurations, according to Proposition 1it corresponds to a finite freezing Boolean network on the grid with only and , or and constant c, v ), we consider three cases.1. If c v = 0, then we can simply remove the vertices v with c v = 0 from the graphsupporting the finite freezing Boolean network dynamics since they are completelypassive (they freeze to 1 if and only if their four neighbors are already fired andfrozen). This construction is done in AC and comes down to deciding if thereis a path from a cell in state 1 to v , which can be done in NL (choose non-deterministically a starting cell in state 1 and travel non-deterministically througha path of length at most nm ). 8 { , , } -FSPP to { , } -FSPP . After one step the grey cells in the macrocell corresponding to a cell withthree grains have three grains (they are neighbor of exactly one cell with four grains).2. If c v = 0 and c u = 0 for all u ∈ N ( v ), then we compute sequentially the answersof the four instances ( c, u ) for u ∈ N ( v ) (still in NL ), and answer positively if andonly if all these four instances are positive.3. If c v = 0 and c u = 0 for at least one u ∈ N ( v ) then we can answer negatively: v needs u to go to state 1 first (strictly before v does), and conversely.Deciding in which of these three cases we are and answering it gives an algorithm in NL for { , , } -FSPP . Proposition 7. { , , } -FSPP ∈ NL . { , , } -FSPP The idea is to reduce the question on a { , , } -simple configuration to a question on a { , } -simple configuration, still on the grid. The transformation is presented on Figure 3,each cell at position ( u x , u y ) ∈ Z of the { , , } -simple configuration is transformed intoa macrocell of size 5 × u x , u y ). The questionedcell is placed on the bottom left corner of the corresponding macrocell (other positionsare possible). This reduction can be computed in constant parallel time, i.e. in AC .The correctness of the reduction is easily deduced from the abelian property of sand-piles (the fact that, when the dynamics converges to a stable configuration, it convergesto the same stable configuration regardless of the order in which firings are performed,in parallel or sequentially [6]). Indeed, if we first consider the firing of values four inthe macrocell corresponding to cell with three grains, then a firing can occur on the { , , } -simple configuration if and only if the whole corresponding macrocell can befired (otherwise, none of the macrocell’s cells is fired, appart from the initialy fired val-ues four in the macrocells corresponding to cells with three grains). The result followsby induction. Proposition 8. { , , } -FSPP ≤ m AC { , } -FSPP , therefore from Propostion 6 wehave { , , } -FSPP ∈ NC . a ∈ { , , , } : a a ∀ a ∈ { , , , } : a a FSPP to { , , , } -FSPP . Top: cells different from v . Bottom: v , with the new questioned cell highlighted. We begin with a trivial remark that
FSPP ≤ m AC { , , , , } -FSPP with the identityfunction since the two problems are identical. We treat subsequent cases one by one,and always use the same reduction technique: a cell of an input for FSPP is convertedto a macrocell ( i.e. a fixed size rectangle of cells) of an input for A -FSPP , in constanttime and in parallel. { , , , } -FSPP The reduction is defined as follows: given an instance ( c, v ) of
FSPP , we replace eachvertex ( u x , u y ) ∈ Z of c with a macrocell of size 5 × u x , u y ). The cell to macrocell correspondence is given on Figure 4. Thisreduction can be computed in constant parallel time, i.e. in AC . Let us denote c ′ theobtained configuration with v ′ the new questioned cell.We now argue in details that ( c, v ) ∈ FSPP if and only if ( c ′ , v ′ ) ∈ { , , , } -FSPP , i.e. the reduction is correct. First, ( c ′ , v ′ ) is a valid instance of { , , , } -FSPP since c ′ is a finite { , , , } -simple configuration. Except for the questioned celland cells without grains (these latter having no influence on the dynamics), there is astrict correspondence between the dynamics of c and c ′ : if vertex ( u x , u y ) of c fires attime t , then vertex (5 u x + 2 , u y + 2) of c ′ fires at time 5 t . Indeed, each backgroundvalue 1 surrounding lines and columns of 3 (which link the centers of macrocells) isneighbor of at most two values 3 (even if we consider the macrocells neighbor to themacrocell corresponding to v when c v = 0), therefore none of them is ever fired andthe correspondence is strict. Regarding the new questioned cell v ′ and its associatedmacrocell, one can simply remark that for any value of c v the new cell v ′ is fired if andonly if 4 − c v centers of neighboring macrocells are fired, and that when this is not (yet)the case then no value 1 in this macrocell is fired. As a consequence we get the result.10 a ∈ { , , , } : a a ∀ a ∈ { , , , } : a a FSPP to { , , , } -FSPP . Top: cells different from v . Bottom: v , with the new questioned cell highlighted. Proposition 9. FSPP ≤ m AC { , , , } -FSPP . { , , , } -FSPP The reduction is defined as follows: given an instance ( c, v ) of
FSPP , we replace eachvertex ( u x , u y ) ∈ Z of c with a macrocell of size 5 × u x , u y ). The cell to macrocell correspondence is given on Figure 5. Thisreduction can be computed in constant parallel time, i.e. in AC . Let us denote c ′ theobtained configuration with v ′ the new questioned cell.The argumentation regarding the correctness of this reduction is analogous to the caseof Proposition 9, except that firings may not be perfectly synchronized because of themacrocell corresponding to the value 1 doing some zigzag, but this has no consequencethanks to the so called abelian property of sandpiles which still holds on freezing sandpiles(any sequence of firings in c is reproduced in c ′ , and conversely). Proposition 10. FSPP ≤ m AC { , , , } -FSPP . { , , , } -FSPP The reduction is defined as follows: given an instance ( c, v ) of
FSPP , we replace eachvertex ( u x , u y ) ∈ Z of c with a macrocell of size 7 × u x , u y ). The cell to macrocell correspondence is given on Figure 6. Thisreduction can be computed in constant parallel time, i.e. in AC . In the constructedmacrocells, each value 2 is neighbor of exactly one value 4, and consequently all becomevalue 3. The rest of the argumentation regarding the correctness of this reduction isanalogous to the case of Proposition 10. Proposition 11. FSPP ≤ m AC { , , , } -FSPP . a ∈ { , , , } : a a FSPP to { , , , } -FSPP . Macrocells corresponding to the questioned cell v are identical, with the newquestioned cell in the center (relative position (3 , { , , } -FSPP We give a reduction from { , , , } -FSPP to { , , } -FSPP , by replacing each vertex( u x , u y ) ∈ Z of c with a macrocell of size 5 × u x , u y ). The cell to macrocell correspondence is given on Figure 7. This reductioncan be computed in constant parallel time, i.e. in AC .The argumentation regarding the correctness of this reduction is analogous to thecase of Proposition 11, with the additional remark that some values 1 in the backgroundmay fire, without any side effect. Since AC is closed by composition, Proposition 11gives the result. Proposition 12. FSPP ≤ m AC { , , } -FSPP . { , , } -FSPP We give a reduction from { , , , } -FSPP to { , , } -FSPP , by replacing each vertex( u x , u y ) ∈ Z of c with a macrocell of size 7 × u x , u y ). The cell to macrocell correspondence is the same as the one given on Figure 6from FSPP to { , , , } -FSPP ), except that the case a = 1 is removed (indeed, remarkthat macrocells do not make use of value 1). This reduction can be computed in constantparallel time, i.e. in AC .The argumentation regarding the correctness of this reduction is analogous to thecase of Proposition 11. Since AC is closed by composition, Proposition 10 gives theresult. Proposition 13. FSPP ≤ m AC { , , } -FSPP . { , , } -FSPP and { , , , } -FSPP Let us first notice that the complexity of predicting both models are equivalent for AC reductions, with the cell to macrocell correspondence given on Figure 8. Proposition 14. { , , , } -FSPP ≤ m AC { , , } -FSPP . a ∈ { , , , } : a a ∀ a ∈ { , , , } : a a { , , , } -FSPP to { , , } -FSPP . Top: cells different from v . Bottom: v , with the new questioned cellhighlighted.0 { , , , } -FSPP to { , , } -FSPP . Left: correspondence when the questioned cell is not a 0, in this casethe new questioned cell is in the center of the corresponding macrocell. Right: if thequestioned cell is a 0 then we inflate all macrocells to be 7 ×
7, and use the picturedmacrocell to replace the questioned cell. 13 Figure 9: Reduction from { , , } -FSPP to strict majority dynamics on a planar undi-rected graph of degree at most 5, for which the complexity of prediction is open.When trying to find an NC algorithm to solve { , , } -FSPP , our attempts to adaptthe reduction to strict majority employed for { , , } -FSPP in Subsection 5.1.2 failed,because gadgets replacing a value 3 seem to require degree five, though they can be madeplanar (see Figure 9, but the case planar of degree at most five is left open by [14, 7]).Remark that we can answer efficiently in many cases using previous developments: • when the questioned cell is a 1 in a cycle of 1, or a 1 on a path whose endpointsare connected to cycles of 1 (decidable in NC ), then the answer is negative (as isthe case for strict majority [14]); • when the questioned cell is a 3 connected via values 3 to a 4 (decidable in NL ),then the answer is positive (as is the case for { , , } -FSPP in Subsection 5.1.4).We conjecture that the remaining cases are equivalent to planar and-or freezing networkswith fan in two and fan out one, but wires are undirected (this is not a circuit) which leadsto difficulties analogous to the general case of { , , , , } -FSPP , though interestinglyin a seemingly more restrictive setting.When trying to prove that FSPP reduces to { , , , } -FSPP , we failed to builda macrocell (with 0 , , ,
4) corresponding to a cell with 2 sand grains (other elementsare straightforward to design), though we found some close constructions. For example,the construction illustrated on Figure 10 behaves almost as a value 2, except that thecombination of west plus east signals does not trigger signals to the north and south (anyother combination of at least two signals triggers signals to the remaining sides). We candeduce that if the number of values 2 is upper bounded by a polylogarithmic functionof the input’s size, then there is an NC truth-table reduction: • for each cell with value 2 we try the macrocell of Figure 10 and the same rotated; • the answer to FSPP will be positive if and only if at least one combination (truth-table) of such macrocells for all values 2 gives a positive answer; • we need to compute the number x of values 2 (in NC ), and then 2 x transformationsin parallel (a polynomial number, each in AC ).We can also use the planar monotone circuit realizing threshold function T (4)2 from [19,Figures 1 or 2] in order to create a macrocell corresponding to a cell with 2 sand grains143 3 3 3 33 3 1 3 3 3 3 1 3 33 4 4 33 33 3 3 3 3 3 3 3 33 3 3 3 3 3 3 3 3 33 1 4 3 3 3 33 3 3 3 4 1 33 3 3 1 4 33 3 3 3 3 3 3 33 4 1 3 3 33 1 4 3 3 3 33 3 3 3 4 1 33 3 3 3 3 3 3 3 3 33 3 3 3 3 3 3 3 33 33 4 4 33 3 1 3 3 3 3 1 3 33 3 3 3 33Figure 10: White cells have no sand grain (0), and diode mechanisms are highlighted.Macrocell with 0 , , , west plus east signalsdo not trigger signals to the north and south sides. Any other combination of at leasttwo signals triggers signals to the remaining sides. The same macrocell rotated by 90degrees is only missing the north plus south combination.(using diodes as on Figure 10), but the result given by the last gate is “trapped” insidethe macrocell (signals are not sent to the remaining sides). We can nevertheless deducefrom such a construction that, if there is only one value 2 and if furthermore this is thequestioned cell, then we have a proper AC reduction. The freezing world allows to make insightful progresses related to difficult questions. Ex-ploiting the formal connections with threshold Boolean functions established by Propo-sition 1, Theorems 1 and 2 characterise the computational complexity of all but tworestrictions of freezing sandpile prediction problem ( FSPP ): either the problem isas hard as unrestricted
FSPP ; or it is proven to be in NC (or below). The results aredisplayed on Table 1. 15 C NL NC FSPP ≤ m AC Open { , } { , , } { , } { , , , } { , , , }{ , , } { , , , } { , , }{ , } { , , , }{ , , } { , , }{ , , } Table 1: Summary of Theorems 1, 2 and Open question 1.The results show interesting fine-grained view on necessary and sufficient conjunc-tions of elements (values among { , , , , } ) for the dynamics to be “as expressive as” FSPP . We propose three remarks.First, in [14] it is proven that the prediction problem on the non-strict majoritycellular automata is P -hard for the family of graphs with maximum degree at least 4, andthat it is in NC for the family of graphs with maximum degree at most 3. In [7] it is proventhat the same problem is in NC when the graph restricted to the two dimensional gridwith von Neumann neighborhood (a particular case of regular graph where each vertexhas degree 4), which corresponds to { , } -FSPP . According to Section 4, the problem { , , } -FSPP introduces a new refinement: when restricted to the two dimensionalgrid with von Neumann neighborhood on which some vertices are somehow removed (with sand value 0), the problem becomes as hard as FSPP .Second, in the reduction of Proposition 11 (and some subsequent ones) it seemsimportant to have many values 4. What is the computational complexity of the weakprediction problem (given a finite stable configuration, plus only one sand grain addition,namely 1 st -col-S-PRED of the survey [4]) in this case?Third, the Open question 1 puts in light a surprisingly complex refinement, whereforbidding only the value 2 seems to decrease the expressiveness of the model, yet notflattening it to another known case. Could it be that, if NC = P , then FSPP and { , , , } -FSPP would belong to different intermediate classes strictly between NC and P (which would exist according to an analog of Ladner’s theorem [26])?The present work circumvents the question of whether FSPP itself is in NC or P -hard. One can implement conjunctions, disjunctions, but the relationship betweenthe impossibility of crossing wires [5] and the possibility of using undirected wires, oreven other forms of signal implementation, leaves open its reduction to MPCVP [27](prediction in NC ), or the possibility to implement non-planar or non-monotone gates [17]( P -hard prediction). The general case of FSPP reduces to { , , } -FSPP , could thathelp in order to find an efficient algorithm? Advances on FSPP would constitute greatinsights for the classical sandpile prediction problem (
SPP ) in two dimensions, left openin the original paper by Moore and Nilsson [21], even though some relationship between
FSPP and
SPP is still to be formally established.Finally, the relationship between threshold functions and cell’s sand content opensperspectives on the prediction of Boolean functions on the grid: in the freezing and16on-freezing worlds, what are the necessary and sufficient elements in order to haveeasy/hard prediction problems?
Acknowledgments
This research was partially supported by ANID via PAI + Convocatoria Nacional Sub-venci´on a la Incorporaci´on en la Academia A˜no 2017 + PAI77170068 (P.M.), FONDE-CYT 11190482 (P.M.), FONDECYT 1200006 (E.G., P.M.), STIC- AmSud CoDANetproject 88881.197456/2018-01 (E.G., P.M., K.P.), ANR-18-CE40-0002 FANs (K.P.).
References [1] P. Bak, C. Tang, and K. Wiesenfeld. Self-organized criticality: An explanation ofthe 1/ f noise. Physical Review Letter , 59:381–384, 1987.[2] E. R. Banks.
Information processing and transmission in cellular automata . PhDthesis, Massachusetts Institute of Technology, 1971.[3] F. Becker, D. Maldonado, N. Ollinger, and G. Theyssier. Universality in FreezingCellular Automata. In
Proceedings of CiE’2018 , volume 10936 of
LNCS , pages50–59, 2018.[4] E. Formenti and K. Perrot. How Hard is it to Predict Sandpiles on Lattices? ASurvey.
Fondamenta Informaticae , 171:189–219, 2019.[5] A. Gajardo and E. Goles. Crossing information in two-dimensional sandpiles.
The-oretical Computer Science , 369(1-3):463–469, 2006.[6] E. Goles, M. Latapy, C. Magnien, M. Morvan, and H. D. Phan. Sandpile modelsand lattices: a comprehensive survey.
Theoretical Computer Science , 322(2):383–407, 2004.[7] E. Goles, D. Maldonado, P. Montealegre, and N. Ollinger. On the computationalcomplexity of the freezing non-strict majority automata. In
Proceedings of AU-TOMATA’2017 , volume 10248 of
LNCS , pages 109–119, 2017.[8] E. Goles and M. Margenstern. Universality of the chip-firing game.
TheoreticalComputer Science , 172(1-2):121–134, 1997.[9] E. Goles and S. Mart´ınez.
Neural and Automata Networks . Springer Netherlands,1990.[10] E. Goles and P. Montealegre. Computational complexity of threshold automatanetworks under different updating schemes.
Theoretical Computer Science , 559:3–19, 2014. 1711] E. Goles and P. Montealegre. The complexity of the majority rule on planar graphs.
Advances in Applied Mathematics , 64:111–123, 2015.[12] E. Goles and P. Montealegre. A fast parallel algorithm for the robust predictionof the two-dimensional strict majority automaton. In
Proceedings of ACRI’2016 ,pages 166–175, 2016.[13] E. Goles, P. Montealegre, K. Perrot, and G. Theyssier. On the complexity of two-dimensional signed majority cellular automata.
Journal of Computer and SystemSciences , 91:1–32, 2017.[14] E. Goles, P. Montealegre-Barba, and I. Todinca. The complexity of the bootstrapingpercolation and other problems.
Theoretical Computer Science , 504:73–82, 2013.[15] E. Goles, N. Ollinger, and G. Theyssier. Introducing Freezing Cellular Automata.In
Proceedings of AUTOMATA’15 , volume 24 of
TUCS Lecture Notes , pages 65–73,2015.[16] R. Greenlaw, H. J. Hoover, and W. L. Ruzzo.
Limits to Parallel Computation:P-Completeness Theory . Oxford University Press, Inc., 1995.[17] R. E. Ladner. The circuit value problem is log space complete for P.
SIGACTNews , 7(1):18–20, 1975.[18] J. Matcha. The computational complexity of pattern formation.
Journal of Statis-tical Physics , 70(3):949–966, 1993.[19] W. F. McColl. On the planar monotone computation of threshold functions. In
Proceedings of STACS’85 , volume 182 of
LNCS , pages 219–230, 1985.[20] C. Moore. Majority-vote cellular automata, ising dynamics, and p-completeness.
Journal of Statistical Physics , 88(3):795–805, 1997.[21] C. Moore and M. Nilsson. The computational complexity of sandpiles.
Journal ofStatistical Physics , 96:205–224, 1999.[22] C. Moore and M. G Nordahl. Predicting lattice gases is p-complete. Technicalreport, Santa Fe Institute Working Paper 97-04-034, 1997.[23] T. Neary and D. Woods. P-completeness of cellular automaton rule 110. In
Pro-ceedings of ICALP’2006 , volume 4051 of
LNCS , pages 132–143, 2006.[24] V.-H. Nguyen and K. Perrot. Any shape can ultimately cross information on two-dimensional abelian sandpile models. In
Proceedings of AUTOMATA’2018 , volume10875 of
LNCS , pages 127–142, 2018.[25] N. Ollinger and G. Theyssier. Freezing, bounded-change and convergent cellularautomata.
Preprint on arXiv:1908.06751 , 2019.1826] H. Vollmer. The gap-language-technique revisited. In
Proceedings of CSL’90 , volume533 of
LNCS , pages 389–399, 1990.[27] H. Yang. An NC algorithm for the general planar monotone circuit value problem.In